PlanetPhysics/Differential Propositional Calculus Appendix 4

Detail of Calculation for the Difference Map
\begin{tabular}{||c||c|c|c|c||} \multicolumn{5}{Detail of Calculation for $$\operatorname{D f = \operatorname{E}f + f$$}} \6pt] \hline\hline & \begin{matrix}{cr} & \operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y} \\ + &                f|_{\operatorname{d}x\ \operatorname{d}y} \\ = & \operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y} \\ \end{matrix} & \begin{matrix}{cr} & \operatorname{E}f|_{\operatorname{d}x\ (\operatorname{d}y)} \\ + &                f|_{\operatorname{d}x\ (\operatorname{d}y)} \\ = & \operatorname{D}f|_{\operatorname{d}x\ (\operatorname{d}y)} \\ \end{matrix} & \begin{matrix}{cr} & \operatorname{E}f|_{(\operatorname{d}x)\ \operatorname{d}y} \\ + &                f|_{(\operatorname{d}x)\ \operatorname{d}y} \\ = & \operatorname{D}f|_{(\operatorname{d}x)\ \operatorname{d}y} \\ \end{matrix} & $$\begin{matrix}{cr} & \operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)} \\ + &                f|_{(\operatorname{d}x)(\operatorname{d}y)} \\ = & \operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)} \\ \end{matrix}$$ \$$6pt] \hline\hline f_{0} & 0 + 0 = 0 & 0 + 0 = 0 & 0 + 0 = 0 & 0 + 0 = 0 \ & \begin{smallmatrix} &  x\ y   & \operatorname{d}x & \operatorname{d}y \\ + & (x)(y)  & \operatorname{d}x & \operatorname{d}y \\ = & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}&\begin{smallmatrix} & x\ (y) & \operatorname{d}x & (\operatorname{d}y) \\ + & (x) (y) & \operatorname{d}x & (\operatorname{d}y) \\ = &    (y) & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}&\begin{smallmatrix} & (x)\ y & (\operatorname{d}x) & \operatorname{d}y \\ + & (x) (y) & (\operatorname{d}x) & \operatorname{d}y \\ = & (x)    & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}&\begin{smallmatrix} & (x)(y) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & (x)(y) & (\operatorname{d}x) & (\operatorname{d}y) \\ = &  0    & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$$ \6pt] \hline $$f_{2}$$ & $$\begin{smallmatrix} & x\ (y) & \operatorname{d}x & \operatorname{d}y \\ + & (x)\ y & \operatorname{d}x & \operatorname{d}y \\ = & (x, y) & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$$ & $$\begin{smallmatrix} & x\  y  & \operatorname{d}x & (\operatorname{d}y) \\ + & (x)\ y & \operatorname{d}x & (\operatorname{d}y) \\ = &     y  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$$ & $$\begin{smallmatrix} & (x) (y) & (\operatorname{d}x) & \operatorname{d}y \\ + & (x)\ y & (\operatorname{d}x) & \operatorname{d}y \\ = & (x)    & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$$ & \begin{smallmatrix} & (x)\ y & (\operatorname{d}x) & (\operatorname{d}y) \\ + & (x)\ y & (\operatorname{d}x) & (\operatorname{d}y) \\ = &   0   & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$$ \ & \begin{smallmatrix} & (x)\ y & \operatorname{d}x & \operatorname{d}y \\ + & x\ (y) & \operatorname{d}x & \operatorname{d}y \\ = & (x, y) & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}&\begin{smallmatrix} & (x) (y) & \operatorname{d}x & (\operatorname{d}y) \\ + & x\ (y) & \operatorname{d}x & (\operatorname{d}y) \\ = &    (y) & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}&\begin{smallmatrix} & x\  y  & (\operatorname{d}x) & \operatorname{d}y \\ + & x\ (y) & (\operatorname{d}x) & \operatorname{d}y \\ = & x      & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}&\begin{smallmatrix} & x\ (y) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & x\ (y) & (\operatorname{d}x) & (\operatorname{d}y) \\ = &  0    & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$$ \6pt] \hline $$f_{8}$$ & $$\begin{smallmatrix} & (x)(y)  & \operatorname{d}x & \operatorname{d}y \\ + &  x\ y   & \operatorname{d}x & \operatorname{d}y \\ = & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$$ & $$\begin{smallmatrix} & (x)\ y & \operatorname{d}x & (\operatorname{d}y) \\ + & x\  y & \operatorname{d}x & (\operatorname{d}y) \\ = &     y & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$$ & $$\begin{smallmatrix} & x\ (y) & (\operatorname{d}x) & \operatorname{d}y \\ + & x\  y  & (\operatorname{d}x) & \operatorname{d}y \\ = & x      & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$$ & \begin{smallmatrix} & x\ y & (\operatorname{d}x) & (\operatorname{d}y) \\ + & x\ y & (\operatorname{d}x) & (\operatorname{d}y) \\ = &  0  & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$$ \ & \begin{smallmatrix} & x  & \operatorname{d}x & \operatorname{d}y \\ + & (x) & \operatorname{d}x & \operatorname{d}y \\ = & 1  & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}&\begin{smallmatrix} & x  & \operatorname{d}x & (\operatorname{d}y) \\ + & (x) & \operatorname{d}x & (\operatorname{d}y) \\ = & 1  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}&\begin{smallmatrix} & (x) & (\operatorname{d}x) & \operatorname{d}y \\ + & (x) & (\operatorname{d}x) & \operatorname{d}y \\ = & 0  & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}&\begin{smallmatrix} & (x) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & (x) & (\operatorname{d}x) & (\operatorname{d}y) \\ = & 0  & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$$ \6pt] \hline $$f_{12}$$ & $$\begin{smallmatrix} & (x) & \operatorname{d}x & \operatorname{d}y \\ + & x  & \operatorname{d}x & \operatorname{d}y \\ = & 1  & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$$ & $$\begin{smallmatrix} & (x) & \operatorname{d}x & (\operatorname{d}y) \\ + & x  & \operatorname{d}x & (\operatorname{d}y) \\ = & 1  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$$ & $$\begin{smallmatrix} & x  & (\operatorname{d}x) & \operatorname{d}y \\ + & x  & (\operatorname{d}x) & \operatorname{d}y \\ = & 0  & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$$ & \begin{smallmatrix} & x & (\operatorname{d}x) & (\operatorname{d}y) \\ + & x & (\operatorname{d}x) & (\operatorname{d}y) \\ = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$$ \ & \begin{smallmatrix} & (x, y) & \operatorname{d}x & \operatorname{d}y \\ + & (x, y) & \operatorname{d}x & \operatorname{d}y \\ = &  0    & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}&\begin{smallmatrix} & ((x, y)) & \operatorname{d}x & (\operatorname{d}y) \\ + & (x, y)  & \operatorname{d}x & (\operatorname{d}y) \\ = &   1     & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}&\begin{smallmatrix} & ((x, y)) & (\operatorname{d}x) & \operatorname{d}y \\ + & (x, y)  & (\operatorname{d}x) & \operatorname{d}y \\ = &   1     & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}&\begin{smallmatrix} & (x, y) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & (x, y) & (\operatorname{d}x) & (\operatorname{d}y) \\ = &  0    & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$$ \6pt] \hline $$f_{9}$$ & $$\begin{smallmatrix} & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\ + & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\ = &   0     & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$$ & $$\begin{smallmatrix} & (x, y)  & \operatorname{d}x & (\operatorname{d}y) \\ + & ((x, y)) & \operatorname{d}x & (\operatorname{d}y) \\ = &   1     & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$$ & $$\begin{smallmatrix} & (x, y)  & (\operatorname{d}x) & \operatorname{d}y \\ + & ((x, y)) & (\operatorname{d}x) & \operatorname{d}y \\ = &   1     & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$$ & \begin{smallmatrix} & ((x, y)) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & ((x, y)) & (\operatorname{d}x) & (\operatorname{d}y) \\ = &   0     & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$$ \ & \begin{smallmatrix} & y  & \operatorname{d}x & \operatorname{d}y \\ + & (y) & \operatorname{d}x & \operatorname{d}y \\ = & 1  & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}&\begin{smallmatrix} & (y) & \operatorname{d}x & (\operatorname{d}y) \\ + & (y) & \operatorname{d}x & (\operatorname{d}y) \\ = & 0  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}&\begin{smallmatrix} & y  & (\operatorname{d}x) & \operatorname{d}y \\ + & (y) & (\operatorname{d}x) & \operatorname{d}y \\ = & 1  & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}&\begin{smallmatrix} & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\ = & 0  & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$$ \6pt] \hline $$f_{10}$$ & $$\begin{smallmatrix} & (y) & \operatorname{d}x & \operatorname{d}y \\ + & y  & \operatorname{d}x & \operatorname{d}y \\ = & 1  & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$$ & $$\begin{smallmatrix} & y  & \operatorname{d}x & (\operatorname{d}y) \\ + & y  & \operatorname{d}x & (\operatorname{d}y) \\ = & 0  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$$ & $$\begin{smallmatrix} & (y) & (\operatorname{d}x) & \operatorname{d}y \\ + & y  & (\operatorname{d}x) & \operatorname{d}y \\ = & 1  & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$$ & \begin{smallmatrix} & y  & (\operatorname{d}x) & (\operatorname{d}y) \\ + & y  & (\operatorname{d}x) & (\operatorname{d}y) \\ = & 0  & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$$ \ & \begin{smallmatrix} & ((x)(y)) & \operatorname{d}x & \operatorname{d}y \\ + & (x\ y)  & \operatorname{d}x & \operatorname{d}y \\ = & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}&\begin{smallmatrix} & ((x)\ y) & \operatorname{d}x & (\operatorname{d}y) \\ + & (x\  y) & \operatorname{d}x & (\operatorname{d}y) \\ = &      y  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}&\begin{smallmatrix} & (x\ (y)) & (\operatorname{d}x) & \operatorname{d}y \\ + & (x\ y)  & (\operatorname{d}x) & \operatorname{d}y \\ = & x       & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}&\begin{smallmatrix} & (x\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & (x\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\ = &  0    & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$$ \6pt] \hline $$f_{11}$$ & $$\begin{smallmatrix} & ((x)\ y) & \operatorname{d}x & \operatorname{d}y \\ + & (x\ (y)) & \operatorname{d}x & \operatorname{d}y \\ = & (x,  y)  & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$$ & $$\begin{smallmatrix} & ((x) (y)) & \operatorname{d}x & (\operatorname{d}y) \\ + & (x\ (y)) & \operatorname{d}x & (\operatorname{d}y) \\ = &     (y)  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$$ & $$\begin{smallmatrix} & (x\ y)  & (\operatorname{d}x) & \operatorname{d}y \\ + & (x\ (y)) & (\operatorname{d}x) & \operatorname{d}y \\ = & x       & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$$ & \begin{smallmatrix} & (x\ (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & (x\ (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\ = &   0     & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$$ \ & \begin{smallmatrix} & (x\ (y)) & \operatorname{d}x & \operatorname{d}y \\ + & ((x)\ y) & \operatorname{d}x & \operatorname{d}y \\ = & (x,  y)  & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}&\begin{smallmatrix} & (x\  y) & \operatorname{d}x & (\operatorname{d}y) \\ + & ((x)\ y) & \operatorname{d}x & (\operatorname{d}y) \\ = &      y  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}&\begin{smallmatrix} & ((x) (y)) & (\operatorname{d}x) & \operatorname{d}y \\ + & ((x)\ y) & (\operatorname{d}x) & \operatorname{d}y \\ = & (x)      & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}&\begin{smallmatrix} & ((x)\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & ((x)\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\ = &    0    & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix}$$ \6pt] \hline $$f_{14}$$ & $$\begin{smallmatrix} & (x\ y)  & \operatorname{d}x & \operatorname{d}y \\ + & ((x)(y)) & \operatorname{d}x & \operatorname{d}y \\ = & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\ \end{smallmatrix}$$ & $$\begin{smallmatrix} & (x\ (y)) & \operatorname{d}x & (\operatorname{d}y) \\ + & ((x) (y)) & \operatorname{d}x & (\operatorname{d}y) \\ = &     (y)  & \operatorname{d}x & (\operatorname{d}y) \\ \end{smallmatrix}$$ & $$\begin{smallmatrix} & ((x)\ y) & (\operatorname{d}x) & \operatorname{d}y \\ + & ((x) (y)) & (\operatorname{d}x) & \operatorname{d}y \\ = & (x)      & (\operatorname{d}x) & \operatorname{d}y \\ \end{smallmatrix}$$ & \begin{smallmatrix} & ((x)(y)) & (\operatorname{d}x) & (\operatorname{d}y) \\ + & ((x)(y)) & (\operatorname{d}x) & (\operatorname{d}y) \\ = &   0     & (\operatorname{d}x) & (\operatorname{d}y) \\ \end{smallmatrix} \6pt] \hline\hline $$f_{15}$$ & $$1 + 1 = 0$$ & $$1 + 1 = 0$$ & $$1 + 1 = 0$$ & $$1 + 1 = 0$$ \6pt] \hline\hline \end{tabular}